Subtracting with number lines

We will use hops on a number line to model subtraction of positive and negative numbers. When subtracting a positive number we move to the left on the number line. When subtracting a negative number we move to the right on the number line. Why is this? Read on...

Example 1: Simplify 5 – 2

To model this problem we use two hops. The first hop is 5 units to the right. Then hop 2 units to the left. Since we end up on 3, the answer is 3.

5 – 2 = 3

Example 5: Simplify (-3) – 2

To model this problem we begin with 3 hops to the left. Subtracting positive 2 means to hops to the left. Since we end up on -5, the answer is -5.

(-3) – 2 = -5

Example 2: Simplify 5 – 8

To model this expression begin by hopping 5 units to the right. Then go 8 units to the left, landing at -3.

5 – 8 = (-3)

Example 6: Simplify (-3) – 8

This expression is model by a hop 3 units to the left, followed by another hop to the left of 8 units.

(-3) – 8 = -11

Example 3: Simplify 5 – (-2)

If subtracting a positive means a move to the left on a number line, then subtracting a negative means a move towards the right. To model this problem, we hop 5 units to the right. The second hop is 2 units to the right (please see note below). Since we end up on 7, the answer is 7.

5 – (-2) = 7

Example 7: Simplify (-3) – (-2)

Just as in Example 3 above, subtracting -2 means moving to the right. So the number line with show two hops: the first is a hop 3 spaces to the left and the second hop 2 spaces to the right. The result is -1.

(-3) – (-2) = -1

Example 4: Simplify 5 – (-8)

To model this problem, go 5 units to the right, and then 8 more units to the right. Since we end up at 13, the answer is 13.

5 – (-8) = 13

Example 8: Simplify (-3) – (-8)

This expression is modeled by hopping 3 units to the left and then 8 units to the right. This lands at 5.

(-3) – (-8) = +5

In the example 5 – (-2) we saw that it was modeled with a 5 unit-hop to the right and then a 2 unit-hop to the right. How can we be certain that subtracting (-2) represents a hop to the right‌?

Let's look at this table of problems and look for a pattern.

5 – 3 = 2

As the second number goes down, the difference gets larger. The pattern verifies that 5 – (-2) is equal to 7. This means subtracting (-2) is accurately represented by a hop to the right.